Weak positive matrices and hyponormal weighted shifts

H. El-Azhar ; K. Idrissi ; E. H. Zerouali

Ufa mathematical journal, Tome 11 (2019) no. 3, pp. 88-98

Résumé

In the paper we study $k$-positive matrices, that is, the class of Hankel matrices, for which the $(k+1)\times(k+1)$-block-matrices are positive semi-definite. This notion is intimately related to a $k$-hyponormal weighted shift and to Stieltjes moment sequences. Using elementary determinant techniques, we prove that for a $k$-positive matrix, a $k\times k$-block-matrix has non zero determinant if and only if all $k\times k$-block matrices have non zero determinant. We provide several applications of our main result. First, we extend the Curto-Stampfly propagation phenomena for for $2$-hyponormal weighted shift $W_\alpha$ stating that if $\alpha_k=\alpha_{k+1}$ for some $n\ge 1$, then for all $n\geq 1, \alpha_n=\alpha_k$, to $k$-hyponormal weighted shifts to higher order. Second, we apply this result to characterize a recursively generated weighted shift. Finally, we study the invariance of $k$-hyponormal weighted shifts under one rank perturbation. A special attention is paid to calculating the invariance interval of $2$-hyponormal weighted shift; here explicit formulae are provided.

Détail

Keywords: subnormal operators, $k$-hyponormal operators, weighted shifts, moment problem.
Mots-clés : $k$-positive matrices, perturbation